Biquaternion Solution of the Problem of Optimal Minimum Time Control of the Solid Spatial Motion

The problem of optimal in the sense of minimum time program control of the spacecraft spatial motion, equivalent to the composition of angular (rotational) and translational (orbital) movements, is investigated. The spacecraft is considered as a free solid of arbitrary dynamic configuration. The boundary conditions for the angular and linear location of the spacecraft, as well as for the angular and linear velocities of the spacecraft are arbitrary. The components of the dual vector function of the optimized control (the dual composition of the vector of the absolute angular acceleration of a solid and a vector equal to the local derivative of the vector of the absolute velocity of the center of mass of the solid (component of the vector of the absolute linear acceleration of the center of mass of the body)) is limited in the its modules. The vectors of the program control force and the program control moment are found in accordance with the concept of solving inverse dynamics problems. To solve the problem, we used the new biquaternion equations of spatial motion of a solid proposed by us where the parabolic Clifford biquaternion (dual quaternion) is used to describe spatial motion. Also the equivalent quaternion equations are used to describe spatial motion with two Hamilton quaternions. Within the framework of the concept of solving inverse problems of dynamics a differential boundary value problem of the twenty-eighth order is obtained using the Pontryagin maximum principle. Examples of numerical solutions of this boundary value problem are given for the cases when the mass distribution of a spacecraft corresponds to a spherically symmetric body and the International Space Station as an arbitrary solid. At the same time, the difference between the initial and final orientations of the spacecraft is large in angular measure and small in linear displacement (we consider the problem of spatial maneuvering). The analysis of the obtained numerical solutions is given.